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The Bogomolny equations are first-order equations obtained by expressing a static energy as nonnegative squares plus a term fixed by the topological charge. Vanishing of the squares minimizes the energy in that topological sector. In particular, the solutions satisfy the full second-order field equations, since their first variation vanishes for compactly supported variations preserving the boundary data. Both monopole and vortex realizations illustrate this mechanism.

For a Yang-Mills theory with gauge group SU(2) and an adjoint Higgs field ${\Phi }^a$, take the static, purely magnetic energyHere $B_i^a=\frac{1}{2}{ϵ}_{ijk}{F}_{jk}^a$, and $e$ is the gauge coupling. In the Bogomolny limit ${\lambda }_H=0$, retain the boundary condition $|\Phi |\to v_H>0$; it breaks SU(2) to U(1) at infinity even though the scalar potential has been removed. The normalized Higgs field on the sphere at infinity defines a map $S_{\infty }^2\to S^2$ of topological degree $n$. In a consistent magnetic orientation the asymptotic flux is $g_m=4\pi n/e$, andFor example, this equality follows by projecting the asymptotic gauge curvature on the Higgs direction: its flux is the integral of the target-sphere area form, divided by $e$.

The gauge-theory Bianchi identity $D_i{B}_i=0$ makes the cross term a divergence:For either sign $s=\pm 1$, complete the square:Choosing $s=\mathrm{sgn}n$ givesThese are the Bogomolny-Prasad-Sommerfield monopole equations. A nonzero scalar potential would add a positive term; a nontrivial smooth monopole with a core cannot generally saturate this same bound at ${\lambda }_H>0$. The Bogomolny monopole equations imply Yang-Mills-Higgs equations: the Bianchi identity already gives $D_i{D}_i\Phi =0$, and the square-completion argument gives stationarity with respect to the gauge field as well.

An explicit charge-one solution is furnished by the hedgehog ansatz for a monopole. To fix its signs, use component conventions $D_i\Phi ={\partial }_i\Phi -e{A}_i\times \Phi$ and $F_{ij}={\partial }_i{A}_j-{\partial }_j{A}_i-e{A}_i\times A_j$. With $\rho =e{v}_{H}r$ and $\hat{x}=x/r$, putThe radial and transverse parts of $D_i\Phi$ are $e{v}_H^2{H}^{\prime }{\hat{x}}_i{\hat{x}}^a$ and $v_{H}HK({\delta }_{ia}-{\hat{x}}_i{\hat{x}}^a)/r$. The corresponding parts of $B_i^a$ are $(1-K^2){\hat{x}}_i{\hat{x}}^a/(e{r}^2)$ and $-v_H{K}^{\prime }({\delta }_{ia}-{\hat{x}}_i{\hat{x}}^a)/r$. Therefore $B=D\Phi$ reduces toThe Prasad-Sommerfield radial monopole solution isDirect differentiation verifies both equations. Near $\rho =0$, $K=1-{\rho }^2/6+O({\rho }^4)$ and $H=\rho /3+O({\rho }^3)$, so the apparent angular singularities disappear and the fields are smooth. At infinity $K$ decays exponentially while $H=1-1/\rho +O(e^{-2\rho })$, leaving an Abelian $1/(e{r}^2)$ magnetic field and mass $4\pi v_H/e$.

Translations of this magnetic monopole give three collective coordinates; a residual U(1) phase provides another in the framed description, where gauge transformations are fixed at infinity. Multimonopole solutions of a common sign also saturate the linear bound. Their widely separated constituents have position and phase parameters, with no static force: the magnetic repulsion and massless-Higgs attraction cancel in this limit. Relative motion is described at low speeds by the moduli-space approximation. An opposite-sign monopole pair does not solve one common sign of the Bogomolny equations and is not protected by that no-force argument.

For the Abelian Higgs model, a second realization occurs in the plane. Let $\psi$ be a complex Higgs field, $D_i={\partial }_i-ie{A}_i$, $B={\partial }_1{A}_2-{\partial }_2{A}_1$, and choose the critically coupled static energyDifferent conventions for the scalar kinetic coefficient change the numerical coefficients of both the bound and the equations. This normalization gives scalar and gauge masses $\sqrt{2}e{v}_H$, explaining the critical balance of their long-distance forces. In three spatial dimensions the planar solutions describe straight strings with this energy per unit length.

Finite energy requires $|\psi |\to v_H$ and $D_i\psi \to 0$. On a large circle the Higgs phase winds by $2\pi n$, so the magnetic flux quantization of an Abelian Higgs vortex givesWriting $J_i=\mathrm{Im}(\bar{\psi }{D}_i\psi )$, expansion and integration by parts giveThe divergence integrates to zero for the usual localized vortex boundary conditions. Thus the Bogomolny square completion for an Abelian Higgs vortex isFor $n\ge 0$ the Bogomolny vortex equations and bound areFor $n<0$, reverse the sign in both first-order equations and obtain $E=2\pi v_H^2|n|$.

For a rotationally symmetric positive vortex number, use $\psi =v_{H}f(r)e^{in\theta }$ and the physical angular component $A_{\theta }=na(r)/(er)$. The Bogomolny vortex equations becomewith $f(0)=a(0)=0$ and $f(\infty )=a(\infty )=1$. The regular core behaves as $f\sim c{r}^n$ and $a\sim e^2{v}_H^2{r}^2/(2n)$. Linearization at infinity gives exponentially decaying massive tails. Unlike the radial monopole, these planar radial profiles have no elementary closed form in general.

The general multi-vortex solution can be encoded by the Taubes equation. Away from zeros, write $\psi =v_H{e}^{h/2+i\chi }$. The first equation givesAt zeros $z_a$ of multiplicity $m_a$, the phase contributes circulation $2\pi m_a$. The second equation becomes, including these Dirac delta terms,For prescribed zeros the planar existence result gives a solution; the prescribed-zero construction of planar Abelian Higgs vortices reconstructs smooth fields, with $h=2m_a\log |x-z_a|+O(1)$ near each zero. Uniqueness is visible directly: for two solutions with the same zeros, their nonsingular difference $w$ satisfies $\Delta w=2e^2{v}_H^2(e^{h_1}-e^{h_2})$. Multiplying by $w$ and integrating giveshence $w=0$. The Abelian Higgs vortex moduli space has arbitrary unordered positions of the $n$ zeros, giving $2n$ real position parameters. The saturated energy is independent of their positions, so critically coupled vortices have no static interaction energy. Away from critical coupling this cancellation is lost.